3.886 \(\int \frac{x^2}{\left (2-3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=47 \[ \frac{4\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}}-\frac{2}{9} x \sqrt [4]{2-3 x^2} \]

[Out]

(-2*x*(2 - 3*x^2)^(1/4))/9 + (4*2^(3/4)*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/(9*
Sqrt[3])

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Rubi [A]  time = 0.0350858, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{4\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}}-\frac{2}{9} x \sqrt [4]{2-3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 - 3*x^2)^(3/4),x]

[Out]

(-2*x*(2 - 3*x^2)^(1/4))/9 + (4*2^(3/4)*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/(9*
Sqrt[3])

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Rubi in Sympy [A]  time = 3.6455, size = 41, normalized size = 0.87 \[ - \frac{2 x \sqrt [4]{- 3 x^{2} + 2}}{9} + \frac{4 \cdot 2^{\frac{3}{4}} \sqrt{3} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-3*x**2+2)**(3/4),x)

[Out]

-2*x*(-3*x**2 + 2)**(1/4)/9 + 4*2**(3/4)*sqrt(3)*elliptic_f(asin(sqrt(6)*x/2)/2,
2)/27

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Mathematica [C]  time = 0.0205282, size = 41, normalized size = 0.87 \[ -\frac{2}{9} x \left (\sqrt [4]{2-3 x^2}-\sqrt [4]{2} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(2 - 3*x^2)^(3/4),x]

[Out]

(-2*x*((2 - 3*x^2)^(1/4) - 2^(1/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (3*x^2)/2]))
/9

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-3*x^2+2)^(3/4),x)

[Out]

int(x^2/(-3*x^2+2)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-3*x^2 + 2)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^2/(-3*x^2 + 2)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-3*x^2 + 2)^(3/4),x, algorithm="fricas")

[Out]

integral(x^2/(-3*x^2 + 2)^(3/4), x)

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Sympy [A]  time = 2.19613, size = 29, normalized size = 0.62 \[ \frac{\sqrt [4]{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-3*x**2+2)**(3/4),x)

[Out]

2**(1/4)*x**3*hyper((3/4, 3/2), (5/2,), 3*x**2*exp_polar(2*I*pi)/2)/6

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-3*x^2 + 2)^(3/4),x, algorithm="giac")

[Out]

integrate(x^2/(-3*x^2 + 2)^(3/4), x)